Thermogravimetric analysis (TGA) is a thermoanalytical technique which measures the change in mass (gains and losses) of a sample as a function of temperature and time. TGA experiments are usually performed by measuring the mass of a sample which is subjected to a temperature program. Known TGA temperature programs can be isothermal, can have a constant heating rate or the heating rate can be related to a function of the mass change. The last temperature program is e.g. realized in TGA instruments by Mettler-Toledo. TGA measurements can provide information about the sample material's properties, such as its thermal stability, as well as its composition.
An approach first introduced in 1969 by Flynn to derive kinetic data from thermogravimetric measurements (J H Flynn, The historical development of applied nonisothermal kinetics, in: Schwenker, R F, Garn, P D (Eds.), Thermal Analysis, Vol. 2, New York: Academic Press; 1969: 1111-1126) was a temperature modulated thermogravimetric analysis (TMTGA) method, which comprised subjecting the sample to a program with a sinusoidal or step-wise temperature change. The temperature program for a TMTGA experiment comprises a temperature with a modulation amplitude Ta and allows to derive kinetic data, such as the average apparent activation energy Eα, from the resulting TMTGA curves. The apparent activation energy Eα of a sample is an important parameter, which is characteristic for said sample, its purity and quality.
Eα can be derived in the temperature interval of the modulation amplitude Ta based on the isoconversion principle, as
                                          E            α                    R                =                                            ln              ⁢                                                          ⁢                              r                ⁡                                  (                                      T                    1                                    )                                                      -                          ln              ⁢                                                          ⁢                              r                ⁡                                  (                                      T                    2                                    )                                                                                        T              2                              -                1                                      -                          T              1                              -                1                                                                        (        1        )            with r:=dα/dt, where Eα is the average apparent activation energy of the conversion α between the temperatures T1 and T2 and r(Ti) is the reaction rate at the temperature Ti. This equation may be considered as being independent of the kind of reaction taking place while the sample is subjected to the temperature program and could therefore be described as being model independent, at least as long as no specific model is selected. The reaction taking place while the sample is subjected to the temperature program will from now on be referred to as “reaction” for ease of reading.
Temperature T1 represents the maximum and temperature T2 the minimum temperature of the temperature modulation. Taking T as the average temperature it follows that T1=T+Ta and T2=T−Ta.
U.S. Pat. No. 6,113,261 A and U.S. Pat. No. 6,336,741 B1 disclose a similar approach to a TMTGA method which utilizes a periodic temperature modulation, such as e.g. a sinusoidal modulation, superimposed on a linear temperature program. The apparent activation energy is here determined as
                              E          α                =                  R          ⁢                                    (                                                                    T                    _                                    2                                -                                  T                  α                  2                                            )                                      2              ⁢                                                          ⁢                              T                a                                              ⁢          ln          ⁢                                    r              ⁡                              (                                  T                  1                                )                                                    r              ⁡                              (                                  T                  2                                )                                                                        (        2        )            with equation (2) being a simple arithmetic rearrangement of Equation (1). A large mass loss or decrease during a single modulation cycle at fast reaction rates generates non-linear effects and additionally numerical errors for the determination of ln r(T1)−ln r(T2). Therefore, the measurements are performed in such a way that the underlying heating due to the linear temperature program can be neglected during a period of modulation, as the temperature modulation amplitude Ta is selected as being so small that the measured signal can be described by linear response theory and be separated by Fourier analysis.
A main drawback of the currently known TMTGA setups is that these are limited to the application of periodic temperature modulations. This becomes particularly evident when applying these periodic TMTGA methods to reactions, which show only low sensitivity and therefore low intensities of the reaction rate r(t) at the beginning or near the end of a reaction. Low intensities in combination with low reaction rates dramatically increase the noise in the numerical results. Further a large mass decrease during a single modulation cycle can occur near the maximum reaction rate, which can cause numerical errors and a lack of data due to a large conversion within one period of the modulated temperature.
Therefore, it would be advantageous to develop a temperature modulated thermogravimetric analysis (TMTGA) method which is more generally applicable and robust with regard to experimental uncertainties and in particular not limited to a periodic temperature modulation.